DNA stretching modeled at the base pair level: Overtwisting and shear instability in elastic linkages Attila Kocsis a , David Swigon b,n a Department of Structural Mechanics, Budapest University of Technology and Economics, Hungary b Department of Mathematics, University of Pittsburgh, United States article info Article history: Received 2 June 2011 Received in revised form 24 October 2011 Accepted 28 October 2011 Available online 12 November 2011 Keywords: DNA mechanics Overstretching Discrete elastic model Simplex algorithm Bifurcations abstract Stretching experiments on single DNA molecules indicate that, counterintuitive to expectations, DNA overwinds when stretched and, at large forces, undergoes a transition into an overstretched form indicated by a plateau on the force–displacement diagrams. It is believed that these effects are the result of non-linearities in the elastic response of DNA. We use a discrete, base pair level model to simulate the behavior of short DNA molecules, taking into account the sequence dependent physical properties of DNA alongside with the coupling between the kinematical step parameters, yet retaining the quadratic form of local elastic energy function. By constructing bifurcation diagrams of equilibrium configurations and studying the dependence on base pair combinations we show that the quadratic model is capable of explaining the overtwisting as a result of coupling between modes of deformation and overstretching as a result of shear instability. & 2011 Elsevier Ltd. All rights reserved. 1. Introduction DNA is a double stranded molecule composed of two poly- nucleotide strands that are bound together by hydrogen bonds between complementary nucleotide bases. In normal conditions the strands wind around the DNA axis as two identical right- handed helices with the diameter of 2 nm and pitch of 3.57 nm. During various important intracellular biological processes, such as transcription or replication, mechanical forces, exerted by various proteins, act on DNA and cause its bending, twisting, stretching, or shearing. Understanding of the role of DNA in such processes requires us to obtain knowledge about the mechanical response of DNA to various loading conditions. Macroscopic properties of DNA can be studied using a range of techniques, including atomic force microscopy (AFM) observations [2], electron microscopy visualization, FRET measurements of dis- tances between parts of a molecule, or single molecule manipula- tion. The latter technique can subject a single DNA molecule to a mechanical loading, consisting of stretching and twisting, by micro- manipulation of objects to which DNA is attached, such as glass needles [3], glass beads [4–6], or magnetic beads [7–9]. Recent DNA stretching experiments [3,5,6,8,10,11] have shown that, contrary to natural intuition with twisted ropes, DNA overwinds under tension until the force reaches a critical value of  30 pN, above that value it unwinds under tension [8]. The magnitude of the overwinding is small, about 2.5 rotations of an 8400 base pair segment [8], which corresponds to about 0.3% of the natural twist of a base pair step. The same effect gives rise to a lengthening of the molecule as a result of overtwisting [8,12]. It has been suggested [8] that one possible explanation of the phenomenon can be obtained by treating the DNA as a micro- structured material consisting of a soft elastic cylindrical core surrounded by stiffer helical backbone threads winding on the surface of the tube. Indeed, when such a structure is stretched, the diameter of the cylindrical core decreases which leads to the overtwisting of the backbone threads which corresponds to the observed phenomena, however the structure maintains the same behavior at all magnitudes of the stretch and cannot therefore be the only explanation of the effect. Its has been also found that when larger forces are applied, DNA undergoes a transition to an overstretched form. During the transi- tion the force remains almost constant while the DNA lengthens to 1.7 times of its natural B-form contour length [5,6,13]. The threshold for this transition is  65 pN, if the molecule is torsionally relaxed, and  110 pN, if it is torsionally constrained. This overstretching transition has been described by two competing models, both of which assume that the plateau is a Maxwell line corresponding to a transition between two phases of the molecule. One model assumes that the new phase is a double-stranded overstretched form, a so-called S-DNA [3], in which the base pair separation is increased but the hydrogen bonding between the bases remains intact. Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/nlm International Journal of Non-Linear Mechanics 0020-7462/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijnonlinmec.2011.10.008 Abbreviation: BVP, boundary value problem; GRS, general representation space; SSA, simplex scanning algorithm; PFA, path-following algorithm. n Corresponding author. E-mail address: swigon@pitt.edu (D. Swigon). International Journal of Non-Linear Mechanics 47 (2012) 639–654